\documentclass[art,10pt]{simple}
\begin{document}
\author{E.~Christopher Lance}
\title{The \texttt{simple} document class and its macros}
\abbrev{simple}
\maketitle
\section{The document class and its options}
The \texttt{simple} document class is a simplified variant of the 
\LaTeX$\,2_\varepsilon$ \texttt{article} class, designed for producing 
straightforward documents formatted for printing on A4 paper in 10, 11 or 
12~point type. Paragraphs are not indented but are separated by a line of 
space (except when the \texttt{art} option described below is in use---then 
paragraphs are formatted as in the \LaTeX\ \texttt{article} style).

Some of the refinements of the \texttt{article} class are not available in
\texttt{simple}. For example, it does not 
support two-sided or two-column options, marginal notes, figures or tables. It 
does allow the standard \LaTeX\ list environments \texttt{enumerate}, 
\texttt{itemize}, \texttt{description}, \texttt{quotation} and \texttt{quote}, 
and these may be nested, but only up to a cumulative depth of two. It also has 
its own quite versatile listing environment \texttt{tab}, which can be used for
constructing most itemised lists. This can be nested inside the \LaTeX\ 
list environments, but not vice versa; and it cannot be nested inside itself. 
It produces a list in which each item has a label (as described below)
followed by an indented first line of text, but the remaining lines of the
item are not indented (beyond the possible existing indentations caused by
one or two levels of \LaTeX\ listings).
 
The format for the \texttt{tab} environment is as follows:
\begin{verbatim}
\begin{tab}<style>
<first item> \\
<second item> \\
...
<last item>
\end{tab},
\end{verbatim}
where \texttt{<style>} is \emph{either} the name of a mathematical symbol 
(\verb0\bullet0, \verb0\circ0, \verb0\ast0, \verb0\star0, \ldots) \emph{or} one 
of the five special commands \verb0\alph0, \verb0\Alph0, \verb0\arabic0, 
\verb0\roman0 or \verb0\Roman0. A list of items is produced, each one prefaced 
by an indentation containing, in the first case, the indicated symbol, and in 
the second case a parenthesised letter or numeral in the indicated style.

The layout of the list can be adjusted by means of the \verb0\tabspec0 command.
This has the form \verb0\tabspec{<specification>}0. Within \verb0<specification>0
you can specify, for example,
\tabspec{\nullspec \labelwidth 1.5em \labelsep 0em}
\begin{tab}\relax
\verb0\labelwidth0, the indentation containing the prefatory symbol flush right, \\
\verb0\labelsep0, the space between the symbol and the text of the item, \\
\verb0\topsep0, the space  at the top and bottom of the list, \\
\verb0\itemsep0, the space between items, \\
\verb0\tabfont0, the font style to be used for the parenthesised letter or 
numeral. 
\end{tab}
You can also use as arguments for \verb0\tabspec0 the predefined settings
\verb0\nullspec0, \verb0\smallspec0, \verb0\medspec0 and \verb0\bigspec0.
These are defined as follows:
\begin{tab}\roman
\verb0\nullspec0: \verb0\topsep0 = zero, \verb0\itemsep0 = zero, \\
\verb0\smallspec0: \verb0\topsep0 = \verb0\smallskipamount0, 
  \verb0\itemsep0 = \verb0\smallskipamount0, \\
\verb0\medspec0: \verb0\topsep0 = \verb0\medskipamount0, 
  \verb0\itemsep0 = \verb0\smallskipamount0, \\
\verb0\bigspec0: \verb0\topsep0 = \verb0\bigskipamount0, 
  \verb0\itemsep0 = \verb0\medskipamount0.
\end{tab}
The default setting is \verb0\smallspec0. For \verb0\labelwidth0 and
\verb0\labelsep0 the defaults are 2.5~em and 0.5~em respectively; for
\verb0\tabfont0 it is \verb0\textit0 when \verb0<style>0 is \verb0\roman0,
and \verb0\textrm0 in all other cases.

For example, the two tabulations above were given by the inputs
\begin{verbatim}
\tabspec{\nullspec \labelwidth 1.5em \labelsep 0em}
\begin{tab}\relax
\verb0\labelwidth0, the indentation containing the prefatory symbol flush right, \\
...
\verb0\tabfont0, the font style to be used for the parenthesised letter or 
numeral. 
\end{tab},

\begin{tab}\roman
\verb0\nullspec0: \verb0\topsep0 = zero, \verb0\itemsep0 = zero, \\
...
\verb0\bigspec0: \verb0\topsep0 = \verb0\bigskipamount0, 
  \verb0\itemsep0 = \verb0\medskipamount0.
\end{tab}.
\end{verbatim}
\gap

The available options for \texttt{simple} are \texttt{10pt}, \texttt{11pt},
\texttt{12pt}, \texttt{art} and \texttt{exam}. Among the first three, the 
default is \texttt{12pt}. For documentation on \texttt{exam}, see a specimen 
examination paper.

The \texttt{art} option provides the format for an article, with numbered 
sections and subsections (but not sub-subsections, paragraphs or sub-paragraphs). 
As in standard \LaTeX, these are introduced by the commands 
\verb0\section{<title>}0 or \verb0\section*{<title>}0 (for an unnumbered 
section), and similarly for subsections. Other standard features are the 
commands \verb0\author{<author>}0, \verb0\title{<title>}0 and \verb0\maketitle0. 

Nonstandard features are \verb0\classno{<class>}0, which inserts a footnote
`1991 \emph{Mathematics Subject Classification}: \verb0<class>0' on the
first page, and \verb0\abbrev{<abbr>}0, which inserts a footnote 
ECL/\verb0<abbr>0 on subsequent pages (for example, \verb0\abbrev{simple}0
occurs in the present document). The command \verb0\seriesnumber{<number>}0
should be used to give the Leeds Department of Pure Mathematics Preprint Series
number in conjunction with the command \verb0\maketitlepage0. This provides a
title page with a display to fit the window in the departmental preprint covers. 
Articles have numbered pages, with the date of printing given as a footnote on
each page, as in the present document. (If the \texttt{art} option is not in
effect then a footnote of the form ECL/\verb0<date>0 can be obtained by
the command \verb0\datestamp0. If in addition numbered pages are required, use 
\verb0\pagenos0 instead of \verb0\datestamp0.)

The environment \texttt{thebibliography}  has an optional
argument \texttt{[<number>]} for which the default is 99. The width of
\texttt{<number>} should be that of the highest numbered item in the
bibliography. If instead of a numeral the argument * is used for \texttt{<number>} 
then the bibliography will be formatted so that items are referred to by the
abbreviation \texttt{<abbr>} rather than numerically. Each item in the 
bibliography is specified by the command
\verb0\bibitem{<abbr>}{\bibname <author>}, <reference>0. For example,

\begin{verbatim}
\begin{thebibliography}[9]
\bibitem{Bla86} 
{\bibname B.~Blackadar}, \emph{K-theory for operator algebras}, MSRI Publ. 5
(Springer, 1986).
%
\bibitem{Lan95} 
{\bibname E.~C.~Lance}, \emph{Hilbert C*-modules: a toolkit for operator
algebraists}, London Math. Soc. Lecture Note Series 210 (CUP, 1995).
%
\bibitem{Wor87} 
{\bibname S.~L.~Woronowicz},  `Compact matrix pseudogroups', 
\emph{Commun. Math. Phys.} 111 (1987) 613--665.
%
\end{thebibliography}
\end{verbatim}
yields
\begin{thebibliography}[9]
\bibitem{Bla86} 
{\bibname B.~Blackadar}, \emph{K-theory for operator algebras}, MSRI Publ. 5
(Springer, 1986).
%
\bibitem{Lan95} 
{\bibname E.~C.~Lance}, \emph{Hilbert C*-modules: a toolkit for operator
algebraists}, London Math. Soc. Lecture Note Series 210 (CUP, 1995).
%
\bibitem{Wor87} 
{\bibname S.~L.~Woronowicz},  `Compact matrix pseudogroups', 
\emph{Commun. Math. Phys.} 111 (1987) 613--665.
%
\end{thebibliography}
\gap

\noindent If the first line of this listing is altered to 
\verb0\begin{thebibliography}[*]0 then the output becomes
\begin{thebibliography}[*]
\bibitem{Bla86} 
{\bibname B.~Blackadar}, \emph{K-theory for operator algebras}, MSRI Publ. 5
(Springer, 1986).
%
\bibitem{Lan95} 
{\bibname E.~C.~Lance}, \emph{Hilbert C*-modules: a toolkit for operator
algebraists}, London Math. Soc. Lecture Note Series 210 (CUP, 1995).
%
\bibitem{Wor87} 
{\bibname S.~L.~Woronowicz},  `Compact matrix pseudogroups', 
\emph{Commun. Math. Phys.} 111 (1987) 613--665.
%
\end{thebibliography}

To cite a reference in the text when using a numerical bibliography, use
\verb0\cite{<abbr>}0. For example, \verb0\cite{Lan95}0 gives \cite{Lan95}.
Multiple references can be given in the form \verb0\cite{Bla86,Lan95,0
\verb0Wor87}0.
Do not use the \verb0\cite0 command when citing references by means of
abbreviations (it should not be necessary anyway, because it is easier to make 
the citation explicitly---just type \verb0[Lan95]0).


\section{The macros package \texttt{\upshape eclmacros.sty}}
This macro file, which is included in the \texttt{simple} document class,
modifies the \LaTeX\ environment for displayed equations in the following ways.
If the first item after the opening \verb0\[0 is \verb0\lineup0 then a set of
aligned equations is obtained, each line being given in the form
\verb0<left side> & <aligned side> \\0. For example,
\begin{verbatim}
\[ \lineup
0 & \leq \< xa-y,xa-y\> \\
  & = a^*\< x,x\> a -\< y,x\> a - a^*\< x,y\> +\< y,y\> \\
  & \leq a^*a -\< y,x\> a - a^*\< x,y\> +\< y,y\> .
\]
\end{verbatim}
will be printed as
\[ \lineup
0 & \leq \< xa-y,xa-y\> \\
  & = a^*\< x,x\> a -\< y,x\> a - a^*\< x,y\> +\< y,y\> \\
  & \leq a^*a -\< y,x\> a - a^*\< x,y\> +\< y,y\> .
\]

\noindent Similarly, if \verb0\bunch0 is the first item after \verb0\[0 then
a set of independently centred displayed equations is obtained, with each line
given as a plain formula ending with \verb0\\0. For example,
\begin{verbatim}
\[ \bunch
(1-2xt+t^2){\partial G\over\partial t} - (x-t)G = 0, \\
t{\partial G\over\partial t} - (x-t){\partial G\over\partial x} = 0, \\
t{\partial\over\partial t}(tG) - (1-tx){\partial G\over\partial t} = 0.
\]
\end{verbatim}
gives
\[ \bunch
(1-2xt+t^2){\partial G\over\partial t} - (x-t)G = 0, \\
t{\partial G\over\partial t} - (x-t){\partial G\over\partial x} = 0, \\
t{\partial\over\partial t}(tG) - (1-tx){\partial G\over\partial t} = 0.
\]

To label a displayed equation (whether single or part of a \verb0\lineup0 or 
\verb0\bunch0), use the command \verb0\tag{<label>}0. For
example, \verb0\[ \<y,x\>\<x,y\>\leq \|\<x,x\>\|\<y,y\> \tag{C-S} \]0 leads to
\[ \<y,x\>\<x,y\>\leq \|\<x,x\>\|\<y,y\> \tag{C-S}. \]
If an equation is so long that it would overlap its label then the label is 
automatically lowered to make room for it, \emph{except} in the case of an
equation in a \verb0\lineup0 display, where the lowering has to be done manually.
This can be achieved through the command \verb0\drop[<dimen>]0, where
\verb0<dimen>0 is an optional dimension whose default value is 
\verb0\baselineskip0. For example, if we add \verb0\drop\tag{longline}0 to the
appropriate line of the above \verb0\lineup0 then we obtain
\[\lineup
0 & \leq \< xa-y,xa-y\> \\
  & = a^*\< x,x\> a -\< y,x\> a - a^*\< x,y\> +\< y,y\>\drop\tag{longline} \\
  & \leq a^*a -\< y,x\> a - a^*\< x,y\> +\< y,y\> .
\]
To refer to a labelled equation, use \verb0\eqnref{<label>}0. Thus
\verb0\eqnref{C-S}0 is printed as \eqnref{C-S}.
\gap

Use the \verb0state0 environment for statements of results. The syntax is
\verb0\begin{state}{<result>}0 for a numbered statement, or
\verb0\begin{state}{<result>}*0 for an unnumbered statement. Example:
\begin{verbatim}
\begin{state}{Theorem} If $A$ is a C*-algebra and $E$ is a countably generated 
Hilbert $A$-module then $E\oplus H_{A} \approx H_{A}\label{absorb}$.
\end{state} 
\begin{state}{Corollary}* If $E$ is a countably generated Hilbert $A$-module 
then $E$ is unitarily equivalent to a fully complemented submodule of $H_{A}$. 
\end{state}   
\end{verbatim}
yields:
\begin{state}{Theorem} If $A$ is a C*-algebra and $E$ is a countably generated 
Hilbert $A$-module then $E\oplus H_{A} \approx H_{A}$\label{absorb}.
\end{state} 
\begin{state}{Corollary}* If $E$ is a countably generated Hilbert $A$-module 
then $E$ is unitarily equivalent to a fully complemented submodule of $H_{A}$. 
\end{state}
Results are numbered sequentially (Lemma 1, Proposition 2, Theorem 3, \ldots);
when the \texttt{art} option is being used, they are numbered within their
section or subsection if appropriate, as in the case of Theorem \ref{absorb}
above. Cross-references can be made as in standard \LaTeX\ by means of the 
commands \verb0\label0 and \verb0\ref0. For example, after including the
command \verb0\label{absorb}0 in the statement of the above Theorem, we can
refer to \verb0Theorem \ref{absorb}0, which will be printed as Theorem 
\ref{absorb}.

For proofs, use the \texttt{proof} environment. This will provide an
italicised heading \emph{Proof}.\ at the beginning and a line of space at the end
of the proof.
\gap

The \texttt{matrix} environment can be used to print matrices of any (reasonable) 
size. Elements of each row are separated by \verb0&0, and rows are delimited
by \verb0\\0. A column can be slightly widened by inserting a ``phantom minus
sign'' \verb0\-0 before one of its entries. Example:
\begin{verbatim}
\[
\begin{matrix}
1&0&\-0&\-0 \\ 4&6&1&0 \\ 4&-1&4&1 \\ -4&0&0&5 
\end{matrix}.
\]
\end{verbatim}
gives
\[
\begin{matrix}
1&0&\-0&\-0 \\ 4&6&1&0 \\ 4&-1&4&1\\-4&0&0&5 
\end{matrix}.
\]
Without the phantom minus signs, this would have unevenly spaced columns:
\[
\begin{matrix}
1&0&0&0 \\ 4&6&1&0 \\ 4&-1&4&1\\-4&0&0&5 
\end{matrix}.
\]
The \texttt{matrix} environment has an optional argument in the form
\verb0[<ldelim>,<rdelim>]0. The default values for the left and right delimiters
are \texttt{(} and \texttt{)}. To obtain a determinant, for example, replace the
second line of the above listing by \verb0\begin{matrix}[|,|]0, giving:
\[
\begin{matrix}[|,|]
1&0&\-0&\-0 \\ 4&6&1&0 \\ 4&-1&4&1\\-4&0&0&5 
\end{matrix}.
\]
Note that for a bracketed matrix, the argument would have to be given in the
form \verb0[[,{]}]0, otherwise the closing bracket would be misinterpreted as the
delimiter for the argument. However, the command \verb0\mx{<data>}0 can be used 
as an abbreviation for \verb0\begin{matrix}[[,{]}]<data>\end{matrix}0. 
Another useful abbreviation is \verb0\ds0 for \verb0\displaystyle0.
\gap

The macro file also contains variants and abbreviations for several mathematical 
symbols. The control sequences \verb0\sint0, \verb0\iint0 and \verb0\iiint0 give
improved spacing for single, double and triple integrals:
\[
\half\sint f(x)\,dx,\qquad \iint f(\vec x)\,d\vec x,\qquad 
\iiint f(\vec x)\,d\vec x.
\]
For definite integrals, use \verb0\defint{<lowerlim>}{<upperlim>}0. For example,
\verb0$\defint{-1}1x\,dx$0 gives $\defint{-1}1x\,dx$.

Small tensor product symbols are given by \verb0\tensor0, \verb0\stensor0 and
\verb0\sstensor0, illustrated in the following expressions with the standard 
\TeX\ symbol \verb0\otimes0 shown below them for comparison:
\[ \bunch
H\tensor K,\qquad x \stensor y,\qquad \theta_{s\sstensor t}, \\
H\otimes K,\qquad x \otimes y,\qquad \theta_{s\otimes t}.
\]
Four commands, \verb0\script0, \verb0\split0, \verb0\german0 and \verb0\vec0 
convert the following letter to a new typeface: calligraphic, `blackboard bold', 
fraktur and boldface respectively. The first two of these also convert the 
letter to uppercase. So \verb0$\script x$0 is $\script x$, \verb0$\split x$0 is
$\split x$, \verb0$\german x$0 is $\german x$ and \verb0$\vec x$0 is $\vec x$.

The abbreviations \verb0\<0 and \verb0\>0 give angled brackets $\<\,,\,\>$. The
fraction $\half$ is given by \verb0\half0 (and it is printed the same size in
a displayed equation, as in the integral further up this page). A centred 
dot given by \verb0\mdot0 differs from the
\TeX\ \verb0\cdot0 by having less space around it. (Compare 
\verb0$(x+y)\mdot(x-y)$0 with \verb0$(x+y)\cdot(x-y)$0:
$(x+y)\mdot(x-y), \ (x+y)\cdot(x-y)$.) The abbreviation \verb0\to0 is used
(as in $\script{ams}\,$\TeX) for the \TeX\ \verb0\rightarrow0 $\to$. A
semidirect product symbol $\sdp$ is given by \verb0\sdp0.

The symbols in the table below are redefined so as to differ from
their appearance in standard \TeX.

\mathchardef\oldepsilon="010F
\mathchardef\oldemptyset="023B
\mathchardef\oldleq="3214
\mathchardef\oldgeq="3215
\begin{center}
\begin{tabular}{c@{\qquad}c@{\qquad}c}
Name&\texttt{simple}&standard \TeX  \\
\hline
\verb0\epsilon0&$\epsilon$&$\oldepsilon$  \\
\verb0\emptyset0&$\emptyset$&$\oldemptyset$  \\
\verb0\leq0&$\leq$&$\oldleq$  \\
\verb0\geq0&$\geq$&$\oldgeq$  \\
\verb0\cong0&$\cong$&\makeatletter
  \def\@vereq#1#2{\lower.5\p@\vbox{\baselineskip\z@skip\lineskip-.5\p@
  \ialign{$\m@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}}$\cong$\makeatother  \\
\verb0\setminus0&$\setminus$&$\backslash$  \\
\end{tabular}
\end{center}
\gap

In plain \TeX, several control sequences of the form \verb0\name0 are used to
ensure that \texttt{name} appears in roman type in formulas (for example,
\verb0\sin0, \verb0\lim0). You can add \texttt{newname} to this list of names
through the command \verb0\rom{newname}0. Thus after \verb0\rom{Ind}0, you can 
type \verb0$\rho' = \Ind_H^G(\rho)$0 to get \rom{Ind}$\rho' = \Ind_H^G(\rho)$. 
If you want there to be a thin space after \verb0newname0, use the command 
\verb0\rom*{newname}0, as in \verb0\rom*{tr}$\tr ab = \tr ba$0, which yields
\rom*{tr}$\tr ab = \tr ba$. (But in that case, do not try to add a suffix to
\verb0newname0, because the space will come first---for example,
\verb0$\tr_X ab$0 will give $\tr_X ab$.)

The command \verb0\pushdown0 lowers subscripts on variables that do not have
superscripts. This can improve the appearance of expressions such as 
$a_ia_i^* = a_i^*a_i$, which is converted to $a_i\pushdown a_i^* = 
a_i^*a_i\pushdown$ by \verb0$a_i\pushdown a_i^* = a_i^*a_i\pushdown$0.

Finally, the useful command \verb0\gap0 at the end of a paragraph inserts an
extra line of space after it.

\end{document}